In what follows, we provide our analysis for (REF ); (REF ) can be analyzed similarly, as explained above.
For simplicity and without loss of generality, we give an exposition of the HP method assuming \(\mu _i(\tau ,x) = \nu _i(\tau ,x) = 0\) , and either \(\sigma _i(\tau ,x) = \sigma _i(\tau )\) , or \(\sigma _i(\tau ,x)\)  is a piecewise constant function of \(x\)  for every layer. In this case each equation \(\mathcal {L}_i u_i = 0\)  by some change of variables \(\tau \mapsto \bar{\tau }, x \mapsto \bar{x}\)  can be transformed to the heat equation (REF ) with \(\sigma ^2_i(\bar{\tau },{\bar{x}}) = 1\)Of course, there exist other possible representations of \(\sigma _i\)  which give rise to the heat equation, or e.g., to the Bessel equation, [1]}., [2]}, and the corresponding Green function \(G(\bar{x},\xi ,\bar{\tau })\)  reads
\( G(\bar{x},\xi ,\bar{\tau }) = \frac{1}{2\sqrt{\pi \bar{\tau }}} e^{-\frac{(\bar{x}-\xi )^2}{4\bar{\tau }}}.\) 
